Problem (REF ) is infeasible for optimal transport between measures \(\mu \)  and \(\nu \)  with \(\Vert \mu \Vert _1 \ne \Vert \nu \Vert _1 \) . This unbalanced case is relevant in the statistical analysis of partial or incomplete data sets.
One remedy is to replace the
hard constraint (REF ) by
a penalty function, e.g. Kullback-Leibler [1]}.
We here follow [2]}, [3]} and present a
linear programming formulation (REF ).
In particular, this formulation can be understood as a moment constrained optimal transport problem.
